Abstract
We construct 4d superconformal field theories (SCFTs) whose Coulomb branches have singular complex structures. This implies, in particular, that their Coulomb branch coordinate rings are not freely generated. Our construction also gives examples of distinct SCFTs which have identical moduli space (Coulomb, Higgs, and mixed branch) geometries. These SCFTs thus provide an interesting arena in which to test the relationship between moduli space geometries and conformal field theory data.We construct these SCFTs by gauging certain discrete global symmetries of mathcal{N} = 4 superYang-Mills (sYM) theories. In the simplest cases, these discrete symmetries are outer automorphisms of the sYM gauge group, and so these theories have lagrangian descriptions as mathcal{N} = 4 sYM theories with disconnected gauge groups.
Highlights
We construct these SCFTs by gauging certain discrete global symmetries of N = 4 superYang-Mills theories
Despite notable recent progress [1,2,3], basic questions about this relationship are unanswered: is a necessary and sufficient condition for an SCFT to have a moduli space that it has a chiral subring? Can the chiral ring have nilpotents? Is the coordinate ring of the moduli space the reduced chiral ring? (I.e., is the moduli space as a complex space given by the set of vevs of the chiral ring fields consistent with the ring relations?) Is the special Kahler structure of Coulomb branches of the moduli spaces encoded in the local operator algebra of the SCFT, and if so, how?
Sclass [4,5,6], geometric engineering [7], and F-theory [8, 9] techniques permit the construction of large classes of Coulomb branch geometries of 4d N = 2 SCFTs
Summary
A key observation of [9] is that at special values of the gauge coupling, certain discrete subgroups, ΣR ⊂ SL(2, Z), of the S-duality group of an N = 4 sYM theory are global symmetries which act non-trivially on the supercharges. The S-duality group is some finite-index subgroup of SL(2, Z) This subgroup can be determined as in [19, 20] by keeping track of the action of SL(2, Z) generators on not just the gauge coupling, and the discrete data specifying the sYM theory. These theories all have Z3, Z4, and Z6 symmetries
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